I too ended up with not enough milk right around 4 months. I know everyone says you *should* be able to make enough, but the fact is many, many women don't (something my pediatrician verified), and you can go crazy trying to up your supply and still not make enough (I've been there, I tried everything, it's just not always fixable, despite what everyone tells you).
I second that you should talk to your pediatrician before trying goat's milk. It's not nutritionally complete.
Two suggestions.
First, you can start introducing solids at 4 months. It's on the early end, but while the recommendation is often to wait till 6 months, it's also not to start before 4 months, so 4 months is old enough. That's how I solved my low-supply issue. A little supplementing with rice cereal, mixed either with water if you have to, or with your own pumped milk.
A second suggestion is to try different formulas. Not sure what you're trying, but many babies can't absorb the milk-based formulas (whey protein is the same as milk protein) or have allergies to the lactose in the milk at that age, so you can either try one where the proteins are broken down (e.g., Infamil Gentlease has broken down proteins and reduced lactose) or you can try soy formula till he's older (I had my son on soy supplement from 4-6.5 months, now he does fine on milk forumla) or there is even a hypoallergenic formula you can try, though it smells nasty. If you went straight to the standard Similac or Infamil, that is likely the problem. They are fairly hard to digest for many babies.
Since he might not like the taste of the formula at first, try mixing the first few bottles with pumped milk (e.g., a few bottles 75% milk, 25% formula, a few 50/50, then 25/75, then 100% forumla).
Good luck!